Correlation of a die's early sum with its total

Question

A fair six-sided die is rolled 60 times. Let S{20} be the sum of the first 20 rolls and S{60} the sum of all 60 rolls. What is the correlation between S{20} and S{60}?

Accepted Answer

Let each roll have variance v (for a die, v = 35/12, but watch how it cancels). Decompose S{60} = S{20} + T, where T (rolls 21 through 60) is independent of S{20}. Then 	ext{Var}(S{20}) = 20v and 	ext{Var}(S{60}) = 60v. Covariance. 	ext{Cov}(S{20}, S{60}) = 	ext{Var}(S{20}) + 	ext{Cov}(S{20}, T) = 20v + 0 = 20v. Correlation. ho = \frac{20v}{\sqrt{20v \cdot 60v}} = \frac{20v}{v\sqrt{1200}} = \frac{20}{\sqrt{1200}} = \sqrt{\frac{400}{1200}} = \sqrt{\frac{1}{3}} \approx 0.577. The per-roll variance v cancels completely; the answer depends only on the shared fraction 20/60 = 1/3. Takeaway.…

Correlation of a die's early sum with its total

Your answer

More Probability questions